The standard model of cosmology is based on the existence of homogeneous
surfaces as the background arena for structure formation. Homogeneity underpins
both general relativistic and modified gravity models and is central to the way
in which we interpret observations of the CMB and the galaxy distribution.
However, homogeneity cannot be directly observed in the galaxy distribution or
CMB, even with perfect observations, since we observe on the past lightcone and
not on spatial surfaces. We can directly observe and test for isotropy, but to
link this to homogeneity, we need to assume the Copernican Principle. First, we
discuss the link between isotropic observations on the past lightcone and
isotropic spacetime geometry: what observations do we need to be isotropic in
order to deduce spacetime isotropy? Second, we discuss what we can say with the
Copernican assumption. The most powerful result is based on the CMB: the
vanishing of the dipole, quadrupole and octupole of the CMB is sufficient to
impose homogeneity. Real observations lead to near-isotropy on large scales -
does this lead to near-homogeneity? There are important partial results, and we
discuss why this remains a difficult open question. Thus we are currently
unable to prove homogeneity of the Universe on large-scales, even with the
Copernican Principle. However we can use observations of galaxies and clusters
to test the Copernican Principle itself.

This interesting paper examines perhaps the most fundamental assumption in cosmology: homogeneity. This seemingly obvious assumption is actually quite difficult to test properly since we can only observe on our past , light cone, not on spatial slices of spacetime. The author examines in detail what observations can test for isotropy, and argues that is far from trivial to prove that the isotropy of the CMB for all observers implies homogeneity.

He also discusses tests of the Copernican Principle, which is the key to linking isotropy we observe to homogeneity. These tests include consistency of luminosity distances (testable by SNIa) and comparison of the radial and transverse BAO signals, as well as tests based on the SZ effect in clusters. In principle these tests can be done well with upcoming dark energy experiments, which I think will be a valuable complement to the traditional dark energy measurements.

These are not really tests of the Copernican Principle i.e. the statement that our position is not special. They are tests of the FRW approximation, i.e. to which degree the FRW model describes the real universe. Even if your position is typical, it may be that your observations are not described by a FRW model because the universe is not exactly homogeneous and isotropic.

True, these tests are all framed as "If we observe such and such, then our universe is non-FLRW." But the idea is to combine them with independent tests of isotropy as discussed in the first half of the paper. Since isotropy+copernican principle -> FLRW, then a universe that fails the FLRW tests but passes the isotropy tests violates the copernican principle.

Observing exact homogeneity and isotropy would prove that the universe is exactly FRW. But this is not the kind of a universe we live in.

Observing statistical homogeneity and isotropy on large scales does not prove that the universe is close to FRW. The FRW model describes an exactly homogeneous and isotropic universe, not one that is statistically homogeneous and isotropic.

Observing exact homogeneity and isotropy would prove that the universe is exactly FRW. But this is not the kind of a universe we live in.

Since that may be too abstract for some people, here is a more concrete way of saying that. The Earth exists, with an overdensity of about 1030. Virialised clusters of galaxies exist, with overdensities above 200. These are uncontroversial observational results. Hence, we do not live in an exactly FLRW universe.

Quote:

The FRW model describes an exactly homogeneous and isotropic universe, not one that is statistically homogeneous and isotropic.